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Reg. No. : .................................... Name : .........................................

III Semester B.Tech. Degree Examination, June 2009 (2003 Scheme) 03 – 301 : ENGINEERING MATHEMATICS – II (CMNPHETARUFB) Time : 3 Hours

Max. Marks : 100 PART – A

Answer all questions.

(10×4=40 Marks)

1. Solve P(p + y) = x(x + y) 2. Find the orthogonal trajectory of the family of parabolas y = ax2. 3. State the Dirichlet’s condition for the convergence of a Fourier series. −π≤ x ≤0 ⎧0, 4. Obtain the Fourier series of f ( x ) = ⎨ ⎩sin x , 0 ≤ x ≤ π π 2 aec θ 2 5. Evaluate ∫ ∫ r cos 2θ dr dθ 0 0

6. Find the angle between the tangents to the curve x = t, y = t2, z = t3 at t = ± 1. 7. Find the directional derivative of u = xy + yz + zx at the point (1, 2, 3) along the X-axis. B

(

)

8. Show that ∫ 2xy + z 3 dx + x 2dy + 3xz 2dz is independent of path joining the A

points A and B.

(

)

9. Evaluate ∫∫ F.nds where F = x + y2 i − 2 xj + 2 yz k and S is the surface of the plane S

2x + y + 2z = 6 in the first octant. 10. Solve (D2 – 4D + 4)y = e2x. P.T.O.

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PART – B Answer one question from each Module.

(3×20=60 Marks)

Module – 1 11. a) Solve P2 + x2 = 4y. b) Solve (D2 – 4D + 4)y = x2 + ex + cos 2x. c) Solve by the method of variation of parameters d2y

12. a) Solve (1 + x )

2

b) Solve

d 2x dt 2

dx

2

+ (1 + x )

− 3x − y = e t ;

d2y dx 2

+ 4 y = sec 2 x

dy + y = 4 cos log(1 + x ) dx

dy − 2x = 0 dt

Module – 2 0 ≤ x ≤1 ⎧ πx , 13. a) Obtain the Fourier series of f ( x ) = ⎨ in the interval (0, 2) ( 2 x ), 1 x 2 π − ≤ ≤ ⎩ 1

b) Evaluate ∫

0

∫

2−x2

x

x x2 + y2

dy dx by changing the order of integration.

0 < x < l /2 ⎩k(l − x), l / 2 < x < l

kx, 14. a) Obtain the half range cosine series for f (x) = ⎧ ⎨

b) Evaluate ∫∫ r 3 dr dθ over the area bounded by r = 2 sin θ and r = 4 sin θ .

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Module – 3 _

−

15. a) Using Stoke’s theorem evaluate ∫ F.d r where F = ( 2 x − y) i − yz 2 j− y 2z k and S c

is the upper half surface of the sphere x2 + y2 + z 2 = 1 and C is its boundary. b) Using Stoke’s theorem prove that div curl F = 0 . ⎛ 1⎞ 16. a) Show that div grad ⎜ r ⎟ = 0. ⎝ ⎠ −

_

b) If F = x 2y j + x 2z k + x 3 i , evaluate ∫∫ F.n ds by using divergence theorem where s

S is the surface bounding the region x2 + y2 = a2; z = 0 and z = b.

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